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 A291717 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry. 10
 1, 4, 6, 9, 36, 8, 16, 120, 24, 168, 25, 300, 72, 714, 178, 36, 630, 144, 2273, 464, 6576, 49, 1176, 288, 5932, 1476, 24288, 6404, 64, 2016, 480, 13536, 3040, 74560, 15680, 341320, 81, 3240, 800, 27860, 6940, 197600, 50860, 1170466, 314862 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 EXAMPLE A configuration of 6 picked points from a 7 X 7 grid with a central (point) symmetry w.r.t. point #, but no line (mirror) symmetry and thus only contributing to T(7,6)=a(27), but not to A291718(27), would be:   o o o X o o o   o o o o o o o   o o o o X o o   o X # X o o o   X o o o o o o   o o o o o o o   o X o o o o o . Triangle begins:    1;    4,    6;    9,   36,   8;   16,  120,  24,   168;   25,  300,  72,   714,  178;   36,  630, 144,  2273,  464,  6576;   49, 1176, 288,  5932, 1476, 24288,  6404;   64, 2016, 480, 13536, 3040, 74560, 15680, 341320; MATHEMATICA decentralize[v_] := 2*Total[v] - Last[v]; T[n_, k_] := decentralize[ Table[ decentralize[ Table[ If[EvenQ[k] || OddQ[a*b], Binomial[ Quotient[a*b, 2], Quotient[k, 2]], 0], {b, 1, n}]], {a, 1, n}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *) PROG (PARI) decentralize(v) = 2*vecsum(v) - v[length(v)]; T(n, k) = decentralize(vector(n, a, decentralize(vector(n, b, if(k%2==0||a*b%2==1, binomial(a*b\2, k\2)))))); for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 16 2017 CROSSREFS Cf. A090642, A098485, A098487, A291716, A291718, A292152, A292153, A292154, A292155, A292156. Sequence in context: A192220 A215477 A292154 * A303699 A292153 A175459 Adjacent sequences:  A291714 A291715 A291716 * A291718 A291719 A291720 KEYWORD nonn,tabl AUTHOR Hugo Pfoertner, Sep 08 2017 EXTENSIONS Terms a(37) and beyond from Andrew Howroyd, Sep 16 2017 STATUS approved

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Last modified June 15 19:31 EDT 2021. Contains 345049 sequences. (Running on oeis4.)